By Fernando Q. Gouvea

The critical subject of this study monograph is the relation among p-adic modular varieties and p-adic Galois representations, and specifically the speculation of deformations of Galois representations lately brought through Mazur. The classical thought of modular types is thought identified to the reader, however the p-adic concept is reviewed intimately, with abundant intuitive and heuristic dialogue, in order that the publication will function a handy aspect of access to analyze in that sector. the implications at the U operator and on Galois representations are new, and may be of curiosity even to the specialists. an inventory of additional difficulties within the box is integrated to lead the newbie in his examine. The e-book will hence be of curiosity to quantity theorists who desire to know about p-adic modular varieties, best them quickly to fascinating learn, and in addition to the experts within the subject.

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These are clearly contained in V, and are they dense in V for every i, for similar reasons. 26 C h a p t e r I. p-adic M o d u l a r Forms Consider first the graded ideal of the graded ring ~ M ( B , k, N) of classical modular forms of level N consisting of the cusp forms, OS(B,k,N) C S. 8, that the image of the ring (~ M ( B , k, N) in V ® B / p B is precisely the subring V1,1. It is then trivial to see that the image of @k~=0S(B, k, N) is precisely vc"'P --1,1 • (Only surjectivity is a problem.

2 Suppose N > 3 and p X N. Then, for any r 6 B such that ord(r) < 1/(p + 1), the homomorphism F r o b : M(B,0, N ; r p) @ K ) M(B,0, N ; r ) @ K is finite and dtale of rank p. 1]. 9 above. 5 below). ~k _ defined by the finite $tale map Frob. Thus, TrFrob(M(B,k,N;r) @ K) C M ( B , k , N ; r p) @ K , so that we have: C o r o l l a r y I I . 3 . 3 Suppose N > 3 and p XN. Then, /or any integer k and any r 6 B such that ord(r) < 1/(p + 1), we have U ( M ( B , k , N ; r ) ® K) C M ( B , k , N ; r p) ® K.

The topology induced from V). Since D' is dense in V, the actions of T and T* extend to V, and the action thus obtained coincides with the one we defined before (because it does so on the dense subspace D'--check on q-expansions). Thus, we can obtain the Hecke operators on V directly from the classical definition. R e m a r k : There is no special reason for using the space D' rather than D, other than the fact that later, when we consider the duality between spaces of modular functions and thqir corresponding Hecke algebras, we will need to exclude the constants.